(5) \( 2 n+5>1 \) and \( 3 n+4>7 \) 6) \( -4 u+9>1 \) and \( 7 u-13 \leq-6 \) (7) \( 32 \leq 3 x+20 \) or \( 17>1-8 x \) 8) \( -2 k+8<14 \) or \( 3 k+1<1 \) 9) \( 5(w+4) \geqslant 5 \) and \( 2(w+4)<12 \)

Sure, let's solve each of the given inequalities step by step. --- ### **5. \( 2n + 5 > 1 \) and \( 3n + 4 > 7 \)** **First Inequality:** \[ 2n + 5 > 1 \\ 2n > 1 - 5 \\ 2n > -4 \\ n > -2 \] **Second Inequality:** \[ 3n + 4 > 7 \\ 3n > 7 - 4 \\ 3n > 3 \\ n > 1 \] **Combined Solution (AND condition):** \[ n > 1 \] *Both conditions must be satisfied, so the solution is \( n > 1 \).* --- ### **6. \( -4u + 9 > 1 \) and \( 7u - 13 \leq -6 \)** **First Inequality:** \[ -4u + 9 > 1 \\ -4u > 1 - 9 \\ -4u > -8 \\ u < 2 \quad \text{(divide by -4 and reverse the inequality)} \] **Second Inequality:** \[ 7u - 13 \leq -6 \\ 7u \leq -6 + 13 \\ 7u \leq 7 \\ u \leq 1 \] **Combined Solution (AND condition):** \[ u \leq 1 \] *Both conditions must be satisfied, so the solution is \( u \leq 1 \).* --- ### **7. \( 32 \leq 3x + 20 \) or \( 17 > 1 - 8x \)** **First Inequality:** \[ 32 \leq 3x + 20 \\ 3x \geq 32 - 20 \\ 3x \geq 12 \\ x \geq 4 \] **Second Inequality:** \[ 17 > 1 - 8x \\ 17 - 1 > -8x \\ 16 > -8x \\ x > -2 \quad \text{(divide by -8 and reverse the inequality)} \] **Combined Solution (OR condition):** \[ x \geq 4 \quad \text{or} \quad x > -2 \] *Simplifying, this covers all real numbers greater than \(-2\). So, the solution is \( x > -2 \).* --- ### **8. \( -2k + 8 < 14 \) or \( 3k + 1 < 1 \)** **First Inequality:** \[ -2k + 8 < 14 \\ -2k < 14 - 8 \\ -2k < 6 \\ k > -3 \quad \text{(divide by -2 and reverse the inequality)} \] **Second Inequality:** \[ 3k + 1 < 1 \\ 3k < 1 - 1 \\ 3k < 0 \\ k < 0 \] **Combined Solution (OR condition):** \[ k > -3 \quad \text{or} \quad k < 0 \] *Since \( k > -3 \) already includes \( k < 0 \) and \( k \geq 0 \), the combined solution covers all real numbers. Therefore, **all real numbers** are solutions.* --- ### **9. \( 5(w + 4) \geq 5 \) and \( 2(w + 4) < 12 \)** **First Inequality:** \[ 5(w + 4) \geq 5 \\ w + 4 \geq 1 \\ w \geq -3 \] **Second Inequality:** \[ 2(w + 4) < 12 \\ w + 4 < 6 \\ w < 2 \] **Combined Solution (AND condition):** \[ -3 \leq w < 2 \] *Both conditions must be satisfied, so the solution is \( -3 \leq w < 2 \).* --- ### **Summary of Solutions:** 1. **Problem 5:** \( n > 1 \) 2. **Problem 6:** \( u \leq 1 \) 3. **Problem 7:** \( x > -2 \) 4. **Problem 8:** All real numbers (any real number satisfies at least one inequality) 5. **Problem 9:** \( -3 \leq w < 2 \) --- If you have any further questions or need additional explanations, feel free to ask!